Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $F:X\to Y$ and $G:Y\to Z$ be functions, with values $F(x)$ as in Functions, Values of a Function, and Functions from One Class to Another §value, let images be as in The Image and the Preimage of a Class under a Class §image and the composite $G\circ F$ as in Relations, Domain, Range, Inverse and Composition §composition, a function from $X$ to $Z$ by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction §composition, and let $A$ and $B$ be subclasses of $X$.

¶ $F[A\cup B]=F[A]\cup F[B]$.

¶ If $A\subseteq B$, then $F[A]\subseteq F[B]$.

¶ $(G\circ F)[A]=G[F[A]]$.

Let now $F$ be injective.

¶ For $x\in X$: $x\in A$ if and only if $F(x)\in F[A]$.

¶ $F[A\cap B]=F[A]\cap F[B]$.

¶ $F[A\setminus B]=F[A]\setminus F[B]$.

¶ $A\subseteq B$ if and only if $F[A]\subseteq F[B]$; and $A=B$ if and only if $F[A]=F[B]$.